Explanatory Questions and Association Targets

From “Does It Matter?” to a Defined Association

An explanatory question asks how an outcome varies across values or categories of a predictor. The word explanatory identifies the question’s purpose: to describe or understand a pattern in the outcome in relation to the predictor. It does not, by itself, say that changing the predictor would change the outcome.

For example, “How does average weekly exercise time vary with age among adults in a specified region?” asks about a pattern in an outcome across values of a predictor. To make the target precise, the question must identify what is measured, for whom, and what feature of the relationship is of interest. A broad question such as “Does age matter?” leaves all of those choices unclear.

Definition: An explanatory question asks how an outcome varies across values or categories of a predictor in a defined target population. An association target is the population quantity that describes this pattern, such as a conditional mean or a contrast between conditional means. An association target describes how variables vary together; it does not, on its own, describe what would happen if someone deliberately changed the predictor.

The observational unit is the entity on which the outcome and predictor are recorded, such as a person, school, or clinic. The target population is the group to which the question refers. Naming these helps prevent a question about one group or unit from being answered using data about another.

Define the Target Before Looking for a Model

For a quantitative outcome \(Y\) and a predictor \(X\), one common association target is the conditional mean function:

m(x)=E(Y\mid X=x)

Here \(m(x)\) is the population mean of the outcome among units whose predictor value is \(x\). It is a population quantity, not the mean observed in one particular sample. The notation \(E(Y\mid X=x)\) refers to an average outcome for the subpopulation at predictor value \(x\); it does not claim that all units in that subpopulation have the same outcome.

The entire function \(m(x)\) describes how the conditional mean varies across predictor values. Sometimes the research question is narrower and asks for a comparison at two particular values. The associated contrast is:

\Delta(x_1,x_0)=m(x_1)-m(x_0)

This contrast is the difference in population mean outcomes between units with predictor value \(x_1\) and units with predictor value \(x_0\). Its units are the units of \(Y\). It is an association between groups defined by \(X\), not necessarily an effect of assigning one predictor value rather than the other.

For a categorical predictor, \(x_0\) and \(x_1\) can be categories, such as “day shift” and “night shift.” For a quantitative predictor, they can be two values, such as 30 and 60 minutes. Choosing two values defines a particular contrast; it does not automatically describe the pattern at every possible value of the predictor.

If the outcome is categorical, a conditional mean may not be the most useful summary. For a binary outcome coded as 1 for “yes” and 0 for “no,” the conditional mean equals the conditional probability of “yes”: \(E(Y\mid X=x)=P(Y=1\mid X=x)\). The research question should make clear which outcome summary is being targeted.

What a Sample Contrast Estimates

In data, the population conditional means are usually unknown. If the sample contains \(n_x\) units with \(X=x\), and their outcome values are \(y_1,\ldots,y_{n_x}\), their sample mean is:

\bar{y}_x=\frac{1}{n_x}\sum_{i=1}^{n_x}y_i

A difference between two such sample means estimates the corresponding population contrast. The arithmetic is direct, but the interpretation must remain tied to the study’s target and data. A sample difference is not automatically a population fact, and neither a sample nor population association alone establishes causation.

The simple identity below shows how the sample calculation corresponds to the population target. Let \(\bar{y}_{x_1}\) and \(\bar{y}_{x_0}\) be the sample means for the two predictor values. Subtracting the sample mean at \(x_0\) from the one at \(x_1\) gives an estimate of the difference between the respective conditional means:

\widehat{\Delta}(x_1,x_0) =\bar{y}_{x_1}-\bar{y}_{x_0} \quad\text{estimates}\quad \Delta(x_1,x_0)=m(x_1)-m(x_0)

This correspondence defines what the calculation is aimed at; it does not prove that the estimate is accurate for the target population. Whether a sample supports generalization depends on how it was obtained and other aspects of study design.

Worked Example: Comparing Two Categories

Suppose a hypothetical sample records weekly exercise time, in minutes, for adults in two age categories. The explanatory question is: “Among adults in the target region, how does mean weekly exercise time differ between adults aged 18–39 and adults aged 40–59?” The outcome is weekly exercise time in minutes; the predictor is age category; the observational unit is an adult. The target is the population difference in conditional means.

In the hypothetical sample, 5 adults aged 18–39 report 90, 120, 150, 180, and 210 minutes. Their mean is:

\bar{y}_{18\text{–}39} =\frac{90+120+150+180+210}{5} =\frac{750}{5} =150\text{ minutes}

For 5 adults aged 40–59, the recorded times are 60, 90, 120, 150, and 180 minutes. Their mean is:

\bar{y}_{40\text{–}59} =\frac{60+90+120+150+180}{5} =\frac{600}{5} =120\text{ minutes}

Taking the first age category minus the second, the estimated contrast is \(150-120=30\) minutes per week. In this hypothetical sample, the adults aged 18–39 averaged 30 more minutes of weekly exercise than the adults aged 40–59. This is a sample association in the specified direction. It does not show that changing a person’s age would change their exercise time.

Choosing a Useful Contrast

A well-framed question says which part of the association matters. Asking about the “relationship” may mean asking whether the conditional mean tends to rise or fall, whether two named categories differ, or whether the means differ at two selected predictor values. These are not identical targets. State the one that matches the substantive question.

For a continuous predictor, two-point contrasts can be useful even when no straight-line relationship is assumed. A contrast between \(x=30\) and \(x=60\) minutes of study describes the difference between those two conditional means. It does not say that the mean changes by the same amount for every additional minute, or that the relationship is linear between or beyond those values.

Worked Example: A Contrast at Two Predictor Values

Consider a hypothetical sample relating daily study time to quiz score, measured in points out of 100. The question is: “Among students in a defined introductory course, how does mean quiz score differ between students who study 30 minutes per day and those who study 60 minutes per day?” The outcome is quiz score in points, the predictor is daily study time in minutes, and the observational unit is a student.

Suppose four sampled students who report 30 minutes of daily study have scores of 68, 72, 75, and 77. Their mean score is:

\bar{y}_{30} =\frac{68+72+75+77}{4} =\frac{292}{4} =73\text{ points}

Suppose four students who report 60 minutes have scores of 74, 78, 80, and 84. Their mean is:

\bar{y}_{60} =\frac{74+78+80+84}{4} =\frac{316}{4} =79\text{ points}

The estimated contrast, 60 minutes minus 30 minutes, is \(79-73=6\) points. In this hypothetical sample, the students reporting 60 minutes of daily study averaged 6 points higher. The result concerns two observed study-time values and an average score. It does not establish that studying 30 additional minutes would raise an individual student’s score by 6 points.

The target should also specify the population. “Among students in this course” and “among all undergraduates” are different questions, even if the outcome, predictor, and contrast are otherwise identical. Likewise, a question about students is not answered by a data set where each unit is a course section and the outcome is the section’s average score.

Turning Vague Wording into an Association Target

Words such as “influence,” “impact,” and “effect” can sound causal even when the available target is only an association. If the intended analysis compares observed groups, a precise question can instead ask how the outcome differs across those groups. The wording should identify the measured predictor values and avoid implying that the comparison represents an intervention.

Worked Example: Making the Population and Outcome Explicit

A vague question is: “Does clinic size affect waiting time?” Rewrite it as an association question and calculate a sample contrast using hypothetical data.

A clearer question is: “Among outpatient clinics in the defined health network, how does mean patient waiting time differ between clinics with fewer than 20 clinicians and clinics with at least 20 clinicians?” The outcome is patient waiting time in minutes; the predictor is clinic-size category; the observational unit is a patient if each patient’s waiting time is recorded. The target is the difference in population mean waiting time between patients attending the two clinic categories. This definition also makes clear that the data must contain patient-level outcomes, rather than only one average per clinic, to support this particular unit-level target.

Suppose a hypothetical sample includes 4 patients at smaller clinics with waiting times of 18, 22, 25, and 27 minutes. Their mean is \((18+22+25+27)/4=92/4=23\) minutes. For 4 patients at larger clinics, suppose the times are 20, 24, 26, and 30 minutes. Their mean is \((20+24+26+30)/4=100/4=25\) minutes. The sample contrast, larger-clinic category minus smaller-clinic category, is \(25-23=2\) minutes.

In this hypothetical sample, patients at clinics in the larger category waited an average of 2 minutes longer. That statement describes the observed association. The word “affect” in the original question would suggest more than this comparison establishes, so it is better not to use it for an association target.

Why the Distinction Matters

An explanatory question can be answered descriptively by summarizing how an outcome varies with a predictor. The same association may be useful for understanding patterns, identifying groups with different average outcomes, or motivating further investigation. But the label “explanatory” does not settle why the pattern occurs. Different groups may also differ in other ways, and the comparison alone does not show that the predictor produced the outcome difference.

A carefully stated association target limits conclusions to what is being compared. It names the outcome and its units, the predictor values or categories, the population, and the population summary. If the target is a conditional mean contrast, its interpretation is a difference in average outcomes for units in those predictor-defined groups—not an individual-level change and not a causal effect.

Key takeaway: State an explanatory association question as a comparison of outcome distributions or summaries across predictor values in a defined population. For a quantitative outcome, a conditional-mean contrast is a precise option. Interpret it as an association unless the study design and analysis justify a causal interpretation.

Common Mistakes

Using causal wording for an association target. “Clinic size increases waiting time” goes beyond a comparison of observed clinic-size groups. Say that mean waiting times differ across the groups unless causal interpretation is justified.
Leaving the comparison unspecified. “How is score related to study time?” does not identify the population summary or predictor values of interest. Specify a trend or a contrast at named values.
Confusing a sample estimate with the population target. A difference between two sample means is an estimate of a population contrast, not the contrast itself.
Leaving out the population or unit. A patient-level question and a clinic-level question concern different observational units and may require different data.
Overinterpreting a two-value contrast. A difference between conditional means at 30 and 60 minutes does not establish a constant change per minute or describe other study-time values.

Check Your Understanding

For each question, identify the association target and describe what a result would—and would not—mean.

Rewrite “Does neighborhood type affect average commute time?” as a question about an association, specifying the outcome, predictor, and target population.
For a quantitative outcome \(Y\), explain in words what \(m(12)=E(Y\mid X=12)\) represents.
A hypothetical sample has a mean outcome of 42 units at \(X=5\) and 47 units at \(X=10\). Calculate the sample contrast for \(X=10\) minus \(X=5\), including its units if the outcome is measured in units, and state its appropriate interpretation.
Why does comparing mean quiz scores for students reporting two different study times not, by itself, show that additional study time caused a score increase?
Give one reason that “among adults in the region” and “among all adults in the country” define different association questions.